{"id":"69d7f956-95d4-4de0-9154-88e0711fd44b","arxiv_id":"2508.18280","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An asymptotic formula for Σ h(a_1γ_1+...+a_mγ_m) over m-tuples of nontrivial zeta zeros with integer coefficients summing to zero, generalizing the Ford-Zaharescu theorem.","lead":"This paper claims an asymptotic formula for sums where several zeros of the Riemann zeta function are combined with integer weights that add to zero, extending the Ford-Zaharescu theorem. The result is useful to number theorists because such sums measure correlations between zeta zeros, quantities central to random matrix predictions about prime distributions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The theorem's hypotheses are undefined in the abstract; without the definition of the admissible class of h and the RH/pair-correlation status, the central asymptotic cannot be assessed.","rationale":"The reader's UNVERDICTED verdict is appropriate. The strongest claim is broad and the abstract lacks the theorem's actual content, so there is no internal mathematical derivation to stress-test. The single load-bearing unknown is the theorem's hypothesis set: the definition of the admissible class of test functions, the conditional/unconditional status, and the uniformity in the weights and multiplicity. These are not idle worries—each one can change whether the asymptotic is genuine, vacuous, or contingent on an unproved conjecture. Because the full text is unavailable, the correct disposition is to leave the reader's UNVERDICTED verdict unchanged while specifying concrete checks that would settle the concern if the text surfaces. This is a missing-evidence finding, not a detected flaw.","tokens_in":885,"tokens_out":1894,"duration_ms":19929,"concrete_test":"Obtain the full text and perform two checks: (1) Inspect the definition of the admissible class of h; verify it contains a natural dense class such as C_c^∞(R) or Schwartz functions, and verify the asymptotic (with explicit error term) for at least one such h with all a_i=±1, comparing numerically with the m=1 Ford–Zaharescu result as T grows. (2) Check whether the theorem states RH or another unproved hypothesis; if it is conditional, re-derive the leading constant from the assumed hypothesis and confirm that the main term equals the Riemann–von Mangoldt density prediction. If the class of h is trivial or the theorem silently assumes RH, the advertised generalization does not land.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is an asymptotic for H over m-tuples of zeta ordinates with integer weights summing to zero. For that claim to be meaningful, the 'special class' of h must support the Fourier inversion that converts the sum into a contour integral over ζ'/ζ; the error term must be controlled uniformly over the weights a_i; and the theorem must state whether it assumes RH, the pair-correlation conjecture, or a zero-density estimate. The abstract withholds all three. Concretely, if the admissible class of h is defined so narrowly that h is supported away from integers/zero or has Fourier transform vanishing on relevant sums, the formula may reduce to a tautology rather than extend Ford–Zaharescu. If the summand repeats zeros by multiplicity while multiplicities are not bounded, an asymptotic for H requires an independent estimate of high multiplicity that is not mentioned. These are not accusations of error; they are unstated premises that carry the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The abstract claims an asymptotic formula for the sum H = Σ_{0<γ_k≤T, 1≤k≤m} h(a_1γ_1+...+a_mγ_m), where the a_i are integers with sum zero, the γ_k run independently over the imaginary parts of the nontrivial zeros of the Riemann zeta function (with multiplicity), and h belongs to an unspecified 'special class.' The abstract states that this generalizes the Ford–Zaharescu theorem from one zero to m-tuples.","tokens_in":992,"tokens_out":4022,"duration_ms":42765,"significance":"If the asymptotic is established for a genuinely broad class of test functions h, the result would be a significant extension of Ford–Zaharescu-type correlation asymptotics to m-point sums over Riemann zeros, with potential applications to the statistical theory of zeta zeros. However, the abstract does not specify the admissible class of h, the form of the error term, the uniformity in the coefficients a_i, or whether the theorem is unconditional or conditional (on RH, a zero-density estimate, or the pair-correlation conjecture). These are load-bearing details for any claim of this type, and their absence prevents an assessment of the significance and correctness of the result.","major_comments":[{"comment":"The phrase 'some special class' (Abstract) is undefined. The admissible class of test functions is load-bearing: if the class is so narrow that h or its Fourier transform vanishes on the relevant integer combinations a·γ, the asymptotic could become a tautology rather than a genuine extension of Ford–Zaharescu. At minimum, the abstract should state the key conditions (smoothness, decay, support of the Fourier transform, or behavior near the origin) or explicitly refer to the theorem in the full text where the class is defined. As written, the claim is not checkable from the material presented.","section":"Abstract"},{"comment":"The abstract does not state whether the theorem is unconditional or conditional. Results of this type typically depend on RH, the pair-correlation conjecture, or a zero-density estimate; the leading constant and the admissible error term depend critically on which input is assumed. Omitting this status makes the strength of the result and the validity of the asymptotic impossible to evaluate. The authors should state, for example, whether the result holds unconditionally for h in a specified class or under RH (or another hypothesis).","section":"Abstract"},{"comment":"The sum counts zeros with multiplicity ('each zero occuring in the sum the number of times of its multiplicity'). If multiplicities are not known to be bounded, a single zero of very high multiplicity could dominate H, and the asymptotic requires control of the multiplicity function (e.g., an upper bound on the number of zeros sharing an ordinate). No such hypothesis appears in the abstract. This is a necessary hypothesis for the claimed asymptotic, not a minor technicality, and should be stated explicitly.","section":"Abstract"}],"minor_comments":[{"comment":"The phrase 'some special class' is unhelpful; use a named class (e.g., Schwartz class, compactly supported, etc.) or give the defining inequalities, or at least point to the section where the class is introduced.","section":"Abstract"},{"comment":"The notation '0<γ_k≤T, 1≤k≤m' is ambiguous. It should be clarified that the sum is over all m-tuples of zero ordinates in (0,T], counted with multiplicity, rather than over independent indices each up to some bound.","section":"Abstract"},{"comment":"'non-trivial' should be 'nontrivial' in formal mathematical writing, and the original Ford–Zaharescu theorem should be cited in the abstract or introduction.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This review is based only on the abstract, which withholds the hypotheses needed to assess the claimed asymptotic: the class of h, the conditional/unconditional status, and the treatment of multiplicities. If the full manuscript supplies a well-defined class and states the necessary assumptions with proofs, the paper may be a meaningful contribution. I would advise the editor to obtain the full text before making a decision; the abstract alone does not support a technical evaluation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The abstract advertises exactly the right kind of next step: a multivariate version of the Ford-Zaharescu theorem, where the sum runs over m-tuples of zeta ordinates weighted by integers summing to zero. If the asymptotic is correct, this is a real extension of the one-statistic setting, and the leading constant being forced by Riemann-von Mangoldt rather than fitted makes circularity a non-issue. Credit where due: the target is a natural and useful object, and the statement as written is not obviously nonsense.\n\nThe soft spot is that the abstract tells me none of the things that carry the theorem. What exactly is the \"special class\" of h? Does the result assume RH, the pair-correlation conjecture, some zero-density estimate, or nothing? Is the error term uniform in the coefficients a_i? The stress-test note lists the right checks: a class narrow enough to make the Fourier inversion a tautology, and multiplicities that need a control independent of the asymptotic. Those are not accusations; they are unstated premises, and I cannot evaluate them from the abstract alone.\n\nI want to be clear that I am not claiming the paper is wrong. I am claiming the abstract underdetermines the mathematics. There is no derivation, no hypothesis list, no error term. For a research announcement that could be fine, but for a referee it means the entire assessment hinges on the theorem statement in the full text.\n\nWho should read this? Specialists in the correlations of zeta zeros, and anyone actively using Ford-Zaharescu-type statistics. If the full paper contains a precise theorem with explicit hypotheses and a proof, it deserves a serious referee. I would not desk reject it on the strength of the abstract. But I would want to see the actual theorem statement before recommending acceptance; the abstract alone cannot carry that weight.","headline":"A promising multivariate extension of Ford-Zaharescu, but the abstract withholds every load-bearing hypothesis, so the whole question sits in the full text.","tokens_in":1555,"tokens_out":1561,"would_cite":false,"duration_ms":20234,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper derives an asymptotic formula for sums over $m$-tuples of Riemann zero ordinates weighted by a function of a zero-sum integer linear combination.","keywords":["Riemann zeta function","nontrivial zeros","asymptotic formula","Ford-Zaharescu theorem","zero correlations","imaginary parts","multiplicity","number theory"],"falsifier":"The abstract withholds both the main term and the class conditions, so a direct test requires the full theorem. With the full formula in hand, one could take $m=2$, $a_1=1$, $a_2=-1$, and $h$ a Gaussian, compute $H$ numerically for increasing $T$, and check whether $H$ divided by the predicted leading factor converges to the stated constant. Any deviation from the predicted limit would refute the claimed asymptotic.","tokens_in":664,"feed_emoji":"🔢","tokens_out":11169,"duration_ms":103421,"temperature":0.7,"pith_summary":"The paper aims to show that a weighted sum over all $m$-tuples of imaginary parts of nontrivial zeros of the Riemann zeta function has a precise limiting form as the upper bound $T$ tends to infinity. Each $m$-tuple is weighted by a function $h$ of the linear combination $a_1\\gamma_1+\\cdots+a_m\\gamma_m$, where the $a_i$ are integers that sum to zero. This is a generalization of the Ford–Zaharescu theorem, which covers the case of a single zero. If the result is correct, it gives the leading-order behavior of $m$-point correlation sums of zeta zeros, a quantity tied to the fine-scale statistics of the zeros.","feed_headline":"Asymptotic formula generalizes Ford–Zaharescu theorem to m zeros","feed_subtitle":"Extends a single-zero theorem to m-tuples of zeta zeros with integer weights summing to zero.","key_machinery":"The central object is the multiple sum $H$ over zero ordinates with weight $h(a_1\\gamma_1+\\cdots+a_m\\gamma_m)$. The condition $\\sum a_i=0$ makes the argument invariant under a common translation of all $\\gamma_i$, so $H$ depends only on the relative configuration of the zeros. The asymptotic evaluation of this translation-invariant sum is the mechanism that carries the argument.","core_discovery":"The paper claims that, for integers $a_1,\\ldots,a_m$ with $\\sum_{i=1}^m a_i=0$ and for every $h$ in a specified 'special class', the sum $$H=\\sum_{0<\\gamma_k\\le T, 1\\le k\\le m} h(a_1\\gamma_1+\\cdots+a_m\\gamma_m),$$ where the $\\gamma_k$ independently run through the ordinates of the nontrivial zeros (each counted with multiplicity), satisfies a derived asymptotic formula as $T\\to\\infty$. The abstract states the sum and the zero-sum condition but does not display the formula's main term or the conditions defining the special class. The result is presented as a generalization of the Ford–Zaharescu theorem from one zero to $m$-tuples.","pith_inferences":["If the theorem is unconditional, it would be a rare higher-order correlation statement about zeta zeros that does not assume the Riemann hypothesis; if it is conditional, the value lies in identifying exactly which unproved input (e.g., a zero-density estimate) suffices.","The content of the result hinges almost entirely on the breadth of the 'special class'; a reader should check whether it includes smooth compactly supported functions or is limited to band-limited test functions, as this determines whether the asymptotic implies genuine distributional statements.","The same zero-sum weighting scheme could in principle be applied to the ordinates of zeros of other $L$-functions or to the zeros of derivatives of $\\zeta(s)$, yielding analogous correlation asymptotics."],"forward_implications":["For $m=1$, the asymptotic formula specializes to the Ford–Zaharescu theorem, giving a direct consistency check of the generalized statement.","For $m\\ge2$, the formula supplies the leading term of the $m$-point correlation sums under zero-sum integer weights, describing how the ordinates of zeta zeros co-vary at large height.","Because zeros are counted with multiplicity, the result applies to repeated zeros as they occur, so it places no restriction on zero simplicity.","The formula holds for every function in the paper's special class, so any function admitted by that class inherits the stated asymptotic."],"supporting_citations":[],"fun_headline_variants":["Zero-sum weighted zeta zeros: new asymptotic formula","Ford-Zaharescu extended to m zeros with zero-sum weights","Asymptotic for sums over weighted zeta zero combinations","New formula for weighted sums of zeta zero ordinates","Generalized Ford-Zaharescu: asymptotic for weighted zero sums"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The theorem's force depends on the unstated definition of the 'special class' of test functions and on any unspoken assumptions about the zeta zeros (such as the Riemann hypothesis or a zero-density estimate); if the class is very narrow or a deep conjecture is assumed, the advertised generalization is much weaker than it appears.","fun_headline_variants_meta":{"raw":{"variants":["Zero-sum weighted zeta zeros: new asymptotic formula","Ford-Zaharescu extended to m zeros with zero-sum weights","Asymptotic for sums over weighted zeta zero combinations","New formula for weighted sums of zeta zero ordinates","Generalized Ford-Zaharescu: asymptotic for weighted zero sums"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000159,"raw_usage":{"total_tokens":1024,"prompt_tokens":661,"completion_tokens":363,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":279}},"tokens_in":405,"tokens_out":363,"duration_ms":4083,"temperature":1.0,"reasoning_tokens":279,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:34:46.392879+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The abstract withholds both the main term and the class conditions, so a direct test requires the full theorem. With the full formula in hand, one could take $m=2$, $a_1=1$, $a_2=-1$, and $h$ a Gaussian, compute $H$ numerically for increasing $T$, and check whether $H$ divided by the predicted leading factor converges to the stated constant. Any deviation from the predicted limit would refute the claimed asymptotic.","supporting_citations":[],"review_version":1}